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,omNAL
OF COMPUTATIONAL
PHYSICS !‘56,201-214
(1993)
Use of a Rotated Riemann Solver for the
Two-Dimensional
Euler Equations
DAVID
Department
W. LEVY, KENNETH G. POWELL, AND BRAM VAN LEER
of Aerospace
Engineering,
University
of Michigan,
Ann Arbor,
Michigan
48109
Received March 4, 1991; revised September 24, 1992
A scheme
for the two-dimensional
Euler equations
that uses flow
parameters
to determine
the direction
for upwind-differencing
is
described.
This approach
respects
the multi-dimensional
nature of the
equations
and reduces
the grid-dependence
of conventional
schemes.
Several
angles
are tested
as the dominant
upwinding
direction,
including
the local flow and velocity-magnitude-gradient
angles. Roe’s
approximate
Riemann
solver is used to calculate
fluxes in the upwind
direction,
as well as for the flux components
normal to the upwinding
direction.
The approach
is first tested for two-dimensional
scalar convection,
where the scheme
is shown
to have accuracy
comparable
to a
high-order
MUSCL
scheme.
Solutions
of the Euler equations
are
calculated
for a variety of test cases. Substantial
improvement
in the
resolution
of shock
and shear waves
is realized.
The approach
is
promising
in that it uses flow
solution
features,
rather
than grid
0 199s
features,
to determine
the orientation
for the solution
method.
Academic Press. Inc.
1. INTRODUCTION
When approximating
hyperbolic
partial-differential
equations in one dimension by upwind differencing, finding
the upwind direction is straightforward.
Characteristic
information can only be transmitted forwards or backwards
depending on the sign of the corresponding characteristic
speed. In two or three dimensions the choice is more
difficult. In the finite-volume formulation of the multidimensional Euler equations, neighboring cells are assumed
to interact through plane waves propagating normal to the
common cell face: the solution to the local Riemann
problem. Upwind-biased fluxes may then be derived just as
for a one-dimensional system of equations. The problem
with this “grid-aligned” approach is that it can misrepresent
the physical features of the flow, unless they happen to lie
along the grid direction. This becomes particularly evident
when computing oblique waves, which tend to become
excessively smeared.
Consider, for example, a pure, steady shear wave oblique
to the grid, as shown in Fig. 1. The component velocity
201
normal to the wave is zero on both sides of the wave-only
the tangential component changes. But in the grid-aligned
frame, the component normal to the ceil face will change.
The solution to the Riemann problem in the grid-aligned
frame will thus include a shock wave or an expansion wave,
depending on the sign of the velocity change. This incorrect
interpretation of the discrete solution in terms of moving
acoustic waves causes the shear wave to spread.
The present research examines the use of a “rotated
Riemann solver,” .m which the upwinding angle is determined not by the grid orientation but by physical features of
the flow problem. The goal is to lessen the dependence of
the solution on grid orientation by exploiting the multidimensional nature of the equations.
Numerical methods based on wave propagation may be
categorized in the following manner:
1. Grid-aligned
methods, in which the fluxes are
calculated in a reference frame aligned with the grid and
cell-centered data are used directly to calculate these. These
methods have found widespread use in practice [I, 21.
2. Rotation methods, in which fluxes are calculated in
coordinates other than a grid-aligned system, but the same
cell-centered data are used to calculate these as in gridaligned methods. The rotation of the coordinate system is
dictated by physical flow features [3-61.
3. Rotation/interpolation
methods, in which fluxes are
calculated in coordinates other than a grid-aligned system,
and an interpolation process is used to estimate data at
desired locations in the flow field, needed for the calculation.
The rotation of the coordinate system and the interpolation
locations are dictated by physical flow features [7-91.
4. Truly multi-dimensional
convection schemes, in
which as little “one-dimensional thinking” as possible is
used. In these methods it is possible for different rotational
frames to be used for different variables [l&12].
Most current schemes fall into the first category, because
it is one of the simplest and most economical implementaOO21-9991/93 $5.00
Copyright 0 1993 by Academic Press, Inc.
All rights of reproduction
in any form reserved.
202
LEVY,
POWELL,
AND VAN LEER
in Powell et al. [ 193 and Roe [20]. Comparative results are
also presented.
The present method belongs to the class of rotation/inter.
polation schemes. In all such schemes, the flow is assumed
to be dominated by one or more waves; a technique is
devised to achieve finite-differencing normal to the wave
front(s) because that results in the best resolution of discontinuities. Implicit in this approach (even if the scheme is
grid-aligned) are three major solution steps:
’
1. Choice of the upwind differencing angle;
2. Determination of a “left” and a “right” state at each
cell interface, as a function of the upwinding angle;
3. Computation of fluxes at the appropriate angle.
FIG.
1.
Pure shear wave oblique to grid.
tions possible. The typical drawback has been discussed
above.
Work on rotation-type schemes includes the solution of
the transonic potential equation in intrinsic coordinates by
Jameson [3]. For the Euler equations, the &scheme
developed by Moretti [ 131 uses certain characteristic directions, while the related QAZlD method of Verhoff et al. [4]
and the method of Goorjian [S, 61 use intrinsic coordinates.
The present work began as an extension of the QAZlD
method. The wave-decomposition scheme of Rumsey et al.
[ 141 uses a fundamentally different Riemann solver for the
flux calculation.
Examples of rotation/interpolation
methods applied to
scalar convection are the “skew upwind scheme” due to
Raithby [7], and the “N-scheme” of Sidilkover [S]. Davis
[9] has implemented a scheme for the Euler equations with
upwind differencing in the frame normal to the shock wave.
A finite-volume scheme similar to the present work is given
by Dadone and Grossman [15].
A method that belongs to the class of truly multi-dimensional schemes is the characteristic interpolation scheme
of Hirsch and Lacer [lo], where certain characteristic
variables are interpolated along associated dominant directions. Other current efforts to develop truly multi-dimensional schemes were made by Powell and Van Leer [ 111
and by Struijs and Deconinck [12]. They consider cellvertex schemes with a downwind-weighted
residual distribution. The diagonalization of Hirsch [16] (same as in
[lo]) for the multi-dimensional Euler equations is used for
wave decomposition, and the convection of each characteristic variable is oriented in the proper direction. A similar
approach, using a different wave model, is described in Roe
Cl73 and has been implemented by De Palma et al. [ 183.
A summary of the wave-decomposition schemes, the truly
multi-dimensional schemes, and the present scheme is given
In the rotation and rotation/interpolation
schemes listed
above, various combinations of upwinding angle and interpolation technique appear. Different upwinding angles are
used, and several sources of input data are tried, ranging
from grid-aligned states to interpolated values. With the
exception of Davis’ scheme [9] and the present scheme,
these schemes compute the flux in a grid-aligned coordinate
system. The attraction
of the rotation/interpolation
approach is that it allows the study of the effect of using a
rotated reference frame, while at the same time standard
Riemann solvers are used. The truly multi-dimensional
schemes require fundamentally new Riemann solvers, which
are not yet mature.
The differences between grid-aligned schemes and the
present scheme based on a rotated Riemann solver all trace
back to differences in the following steps:
1. In a grid-aligned scheme, the direction of upwind differencing is normal to the cell face. In the present scheme,
the upwinding angle can be freely chosen, and, in particular,
can be based on features such as flow and pressure-gradient
angles.
2. In a first-order grid-aligned scheme, the left and right
states are taken to be the average quantities in the cells on
either side of the cell face. In the present scheme, the left and
right state vectors depend continuously on the upwinding
direction. The first-order scheme uses interpolation between
cell centers to obtain these states.
3. In a grid-aligned scheme, the flux normal to the
upwinding direction does not contribute to the flux through
the cell face, so it is not calculated. In the present scheme;
the fluxes at the chosen angle, as well as in the direction
normal to it, must be calculated. The flux normal to the-cell
face is computed by projecting these two flux components
onto the coordinate frame aligned with the cell face.
”
Before discussing the formulation for the Euler equation%
these considerations are examined in the context of the twO?
dimensional scalar convection equation (Section 2). It is
then possible to analyze several aspects of the schemei
ROTATED
RIEMANN
including monotonicity. The scheme is then extended to
the two-dimensional Euler equations (Section 3). Various
upwinding angles are tested, all using the approximate
Riemann solver due to Roe [Zl ]. The lessons learned from
the monotonicity analysis of the scalar equation are then
applied to the Euler equations. Results for three different
flow problems are shown (Section 4). Conclusions and
recommendations are presented in Section 5.
2. SCALAR
CON+ECTION
In the development of schemes to model equations that
describe convection, it is useful to apply these schemes first
to the simple convection equation. In two dimensions, the
convection equation is
*
u,+au,+bu,=O.
(1)
In general, the coefficients a and b may be functions of x, y,
or, in the nonlinear case, u.
Formulation of the Grid-Aligned Scheme. A first-order
accurate, grid-aligned upwind method is formulated by
calculating fluxes based on the upwind, cell-centered state
values. For example (if a is constant),
a>0
a < 0.
(2)
A better approximation method for the fluxes (although
still grid-aligned) is the MUSCL approach, due to Van Leer
[22,2]. For example, with a > 0, the flux in Eq. (2) can be
computed with
Ui-
1/2,j=Ui-l,j+
i
i[(l-KS)d~+(l+Ks)d:]
;
1 i-
l,j
(3)
the forward and backward difference operators are defined
by, respectively,
203
SOLVER
regions of uniform flow, where the differences are very small.
The value E = 1 x lo-” is typically used for double precision calculations.
Formulation of the Characteristic Interpolation Scheme.
When the requirement of using grid geometry in the determination of cell face states is relaxed, it becomes apparent
that a large class of schemes can be designed in which the
cell-face state values are traced back to some point within
the domain of dependence for that face. The first step in the
specification of the cell-face state is to select a direction for
upwind-differencing;
the convection angle is a logical
choice. The other important direction to consider is that of
the solution gradient. However, in the linear case, the steady
convection and solution gradient angles are orthogonal, so
these define the same orthogonal frame. In the scalar case,
the convection angle defines the characteristic direction, so
this method is referred to as “characteristic interpolation.”
Once the upwinding angle has been specified, there is still
a large range of possibilities for interpolation locations. To
simplify the interpolation process, the state locations are
chosen to be on lines connecting adjacent cell centers.
For example, when a and b are both constants and
0.5 < b/a < 2.0, the interpolation locations for all four faces
of a cell are shown in Fig. 2. There are more possibilities,
however, in the case of arbitrary convection speeds. Then,
a complete template of cells surrounding the cell face of
interest must be considered. Linear interpolation of states
between cell centers is used,
u*=(l-w)u,+wu,,
(6)
where u* is the interpolated state, w is the linear interpolation weight, and u, and u2 are the two states located at the
cell centers which straddle the characteristic line. For cases
where a or b are zero, the linear interpolation gives the cell
center state and the characteristic interpolation scheme
reduces to the grid-aligned scheme.
YIAY
0.5 < b/a < 2.0
I
(di+)i,j=Ui+l,j-Ui,j3
(Al)i,j=Ui,j-Ui-I,j*
(4)
With K = 4, this corresponds to parabolic interpolation with
the stencil centered on (i- 1,j). Also included in Eq. (3) is
the limiter function s = s(d + , d _ ), which is used to prevent
overshoots and undershoots near discontinuities. The
differentiable limiter due to Van Albada is used:
2A+A-+E
S(A+‘A-)=(A+)2+(A-)2+EThe parameter
E is used to prevent division by zero in
j
-
j-l -
t
i-l
i
x/AX
FIG. 2. Locations for interpolation of statequantities.
LEVY,
204
POWELL,
AND
The interpolated state values are used as approximations
for the cell face states for the flux computation, e.g.,
Fi-
l/2, j
= au,*_ l/2,
VAN
LEER
lines for the case a > b > 0. The flux formulas obtained from
characteristic interpolation for the north and south faces
are:
(7)
j-
Fjv=[(b-;)
u,+;u,]
(lla)
Analysis of the Characteristic Interpolation Scheme. For
the case in which 0.5 6 b/a < 2, shown in Fig. 2, the forward
Euler update for cell 0 is then dependent only on cells &3
and may be written
1.
+:(a-b)u;-i(a+b)u;
L
L
(8)
J
This is equivalent to a cell vertex scheme, in which the
residual for the imaginary cell 0123, centered at vertex
(i - 4, j - i), is calculated using the trapezoidal rule for the
fluxes. The residual is then assigned fully to the downwind
node, node 0. The scheme will be second-order accurate in
space, but will not be monotone, as shown below.
This level of accuracy is perhaps unexpected, because the
characteristic upwinding scheme, from a geometric peispective, would appear to be only first-order accurate. Data used
for the face fluxes are taken some distance away from the
face itself. However, since the solution of the equation is
constant along characteristic lines, the first-order error in
the extrapolation to the cell face disappears in the steady
state.
The proper& of positivity can be examined by writing the
scheme in the form
n+lUO
-
j,
(9)
WI?
where the sum is over all cells which contribute to the
residual. A scheme will be monotone if none of the coefficients ck is negative (see Godunov [23]). Rewriting
Eq. (8) in this form reveals:
CO= 1 - $ (v, + v,),
F,=[(b-;)]ug+;u,].
Sidilkover observed that modifying the weights in these
formulas to obtain positivity can be regarded as the result of
limiting the angle by which each characteristic deviates from
the cell-face normal. If the upwinding angle is restricted to
7rn/4for these two faces, then the fluxes become
FN = +b(u,, + ul)
F, = $b(ug + u,),
and the coefficients that result are
c,=l-v
a, Cl = v, - Vb,
$b
-
c,=o.
(13)
......
.
p----”
.:::::,,
;::::.
..........
........
......
....
.::.......
......
........
..........
............
0......
d_
E
v,),
where v, = a d t/h and vb = b d t/h are the Courant numbers
in the x and y directions. It can be seen that co > 0 if v, < 1
and ttb < 1. The coefficient c2 is always positive, but c1 and
c3 cannot be positive simultaneously. They can both be
zero, when v, = vb.
In [S], Sidilkover derives a monotone scheme, termed by
him the “N-scheme.” It can be interpreted in a geometric
manner as a restriction on the interpolation procedure used
to find the upwind states. Figure 2 shows the characteristic
vb,
Results and Discussion. The first-order grid-aligned,
K = f grid-aligned, characteristic interpolation, and limited
(restricted) interpolation schemes are tested on two scalar
convection problems, both on the domain 0 6 x d 1 and
0 < y < 1. Both problems may be described as “circular
convection,” where the convection speeds are defined by
a = a(y) = y and b = b(x) = i - x. Particles should follow
cl = +(va - vb),
c3 =
c2 =
Since a > b > 0, this is a positive scheme. A similar restriction on the interpolation for the vertical faces works for the
case b > a. The full range of monotone upwinding angles is
shown in Fig. 3.
(10)
C2=fh~+~b),
’ (llb)
Vertical
Faces
FIG. 3. Limited
teristic interpolation
Horizontal
domain
scheme.
of dependence
for
Faces
the monotone
charac-
ROTATED
RlEMANN
clockwise circular paths about the location ($, 0). In the
first case, the lower inflow boundary is split up as follows:
O<x<O.125,
U= 0
0.125 <x-c 0.375,
u=l
0.375 <xc
u = 0.
0.50,
(14)
This describes a “tophat” shape, and will test the four
schemes’ performance at resolving discontinuities. The
second case uses a Gaussian inflow specification at the
lower left boundary:
@, 0) = e-W”--.25)2b
(15)
This case will result in a smooth solution, providing a better
measure of the schemes’ overall accuracy. In each case, the
left, upper, and right inflow boundaries are initialized to the
exact solution.
Using a 64 x 64 cell Cartesian grid, steady-state solutions
are computed. Profiles along the lower boundary for the
tophat circular convection are shown in Fig. 4. The data
shown are cell vertex values, obtained by postprocessing the
cell-centered results. The profiles for 0~ ~~0.5 show the
input conditions, while the profiles for 0.5 <x < 1.0 show
-----------
1st Order, Grid-Aligned
~=1/3, Grid-Aligned
FWI Characteristic
United
Characteristic
205
SOLVER
the computed solution after convection through the
domain.
As expected, the first-order grid-aligned solution is considerably diffused, and the K = $ grid-aligned scheme gives
substantial improvement. The full, or unlimited, characteristic interpolation scheme results in the sharpest of the
discontinuities, but at the expense of monotonicity. This is
predicted by the positivity analysis above. The angle limiting procedure for the characteristic interpolation scheme
works well, giving results comparable to the K = f gridaligned scheme. The profiles for the Gaussian case, Fig. 5,
show similar trends. In this case, the full characteristic
upwinding method preserves the input profile without
apparent change.
For this problem, the exact solution is known, so a
precise evaluation of the global error is possible. One
measure of this is the L, error norm calculated as
112
L2=
$,j
[
(uq-%,jJ2]
r,,=
(16)
.
1
The cell average computed and exact solutions are used to
calculate the error. The discrete and exact solutions are
calculated for the Gaussian case on 16*, 322, 642, and 962
cell Cartesian grids, and the error norms are plotted in
Fig. 6. The relative accuracies of the schemes are shown to
be the same as indicated by the profile figures. The order
of accuracy of the schemes is given by the slopes of the
-----------
1st Order, Grid-Aligned
rr=l/3,
Grid-Aligmd
Fbll Characteristic
Liited
Characteristic
4
O.W-
0.70u
0.50-
0.30-
O.lOJ
FIG. 4. Tophat
boundary,
computed
circular
convection
cross
on a 64 x 64 cell grid.
section
along
the
lower
FIG. 5.
boundary,
Gaussian
circular
convection
computed on a 64 x 64 cell grid.
cross
section
along
the lower
LEVY, POWELL,
206
o-----4
-
present work investigates the resulting improvement when a
single, dominant, upwinding angle is used. The conventional Riemann problem, which in grid-aligned methods is
solved in a frame normal to the face of a computational cell,
is now solved in a frame rotated about the cell face midpoint. This class of scheme will be called “rotated Riemann
solvers.” In this approach, it is possible to isolate the effects
of differing reference frames. In addition, standard flux formulae may be used, which have been in use for some time.
The truly multi-dimensional schemes referred to earlier are
not as mature and are not as modular in nature.
In formulating the rotated-Riemann-solver
method, the
major points considered are:
1st Order, Grid-Aligned
r(=1/3,
Grid-Aligned
char~cristic
-alI
--Limited
characteriltic
0.00-l
-l.OO-
J52)
-2.lm-
1. Choice of upwind angle;
2. Determination of a left and a right state at each cell
interface, as a function of the upwind angle;
3. Computation of fluxes at an appropriate angle.
-3.00-
-4.00
AND VAN LEER
,
-2.00
I
-1.50
I
1
-1.00
log@)
FIG. 6. Error norms for the Gaussian circular convection problem.
log(&) error norms. These are calculated,
regression, to be
First-order, grid-aligned
0.607
K = f, grid-aligned
1.454
Full characteristic
1.879
Limited characteristic
0.795
using linear
It is seen that the limited characteristic scheme behaves
more as the first-order scheme, but with a reduced error
coefficient, while the full characteristic scheme mimics the
convergence of the K = 4 grid-aligned scheme, again with
reduced error.
Overall, the MUSCL scheme produces the best quality
results, although the limited characteristic interpolation
scheme is a close second. Both of these schemes use
interpolation
methods similar in complexity, but the
characteristic interpolation scheme uses a more compact
stencil.
3. FORMULATION
OF THE ROTATED
RIEMANN
SOLVER METHOD
FOR
THE EULER EQUATIONS
In this section, the characteristic interpolation method for
scalar convection is adapted to the Euler equations.
Although multiple waves may exist locally in the solution-each with its own optimum upwind direction-the
The performances of the grid-aligned and rotated-Riemannsolver schemes are then compared for channel flows and
flows about airfoils in Section 4.
Angles for Upwind Differencing.
Since there is more
than one characteristic direction for the multi-dimensional
Euler equations, a choice must be made. The following
choices are considered:
1. Flow angle;
2. Pressure-gradient direction;
3. Velocity-magnitude-gradient
direction.
Perhaps the most basic feature in a flow field is the
streamline angle, so it is natural to consider it as the
dominant direction. This frame is chosen several times in
previous work..discussed above. The local flow angle at a cell
face is computed from the velocities in the neighboring cells
L and R:
0,=tan-’
s
(
.
L
R
)
(17)
To align the Riemann solver with a shock wave, the
pressure-gradient angle can be used. The pressure-gradient
field is more difficult to compute than the Bow-angle field,
but resolution of shock waves should improve if this
orientation is used. To compute the pressure-gradient angle,
Green’s theorem is applied to the scalar pressure, p,
jl
Vpds=f
V
pfidl,
f-W
CiV
where fi is the unit normal vector, positive outward. The
average pressure-gradient is then
pi3 dl,
(19)
ROTATED
RIEMANN
where the integration is in the counterclockwise direction.
The average pressure gradient at a cell face can then be
calculated using an appropriate template of cells surrounding that cell face.
It may also be desirable to align the rotated Riemann
solver with pure shear waves, which have no pressure
change. In this case, the gradient of the velocity magnitude
can be used:
vq=vJzTz
(20)
This “q-gradient” can be calculated in exactly the same
manner as the pressure gradient. It has the added benefit of
sensing shock waves as well. In the presence of a pure shock
wave, pressure-gradient and q-gradient upwinding will give
identical results.
.
Interpolation Methods for State Quantities. Once the
upwinding direction has been determined, the next step is to
determine the state quantities used as inputs to the flux
function. An interpolation method is used to approximate
the states at a point on the line determined by the upwinding direction and the cell face midpoint. Figure 7 shows
a template of six cells surrounding a vertical cell face. As
with the characteristic interpolation method used for the
scalar convection equation, linear interpolation is used !to
approximate the states at the point of intersection of the
upwinding line and the line connecting the two cell centers
that straddle the upwinding line. In the work of Dadone and
Grossman [ 151, the cell average value corresponding to the
cell nearest the upwinding line is used. For the Euler equaEl
A
+
m
Dominant Left
Dominant Flight
Minor Left
Minor Right
tions, four locations are required for interpolated states: left
and right states for the upwind direction, and left and right
states in the normal direction. The states which lie along the
upwind line will be called the “dominant” states, while the
states which lie on the normal line will be called the “minor”
states. A robust searching algorithm, described in detail in
[24], is used to identify the locations used for interpolation
on nonuniform grids.
With this six-cell template required to calculate the flux
through a cell face, a total of nine cells, shown in Fig. 8, will
be used to update the cell averaged state values. The cell
update template used for the MUSCL scheme is also shown.
Whereas the same number of cells are used for each, the
rotated-Riemann-solver
template is more compact.
Flux Formulation.
7.
Interpolation templatefor the rotated Riemannsolver.
The generic formula for a flux normal
to a cell face is
F = W-J,, Ud;
(21)
in a conventional first-order scheme the left and right input
states simply are the average values in the adjacent cells.
Convection speeds are then based on velocity components
parallel and perpendicular to the cell face; the upwinding
direction, i.e., the direction in which the Riemann problem
is solved, is normal to the cell face.
When an upwind direction, 8, is chosen that is not normal
to the cell face, the flux component in this direction is
F, = b(U~,,
&,I,
(22)
where the subscript D indicates the input states are found
in the upwinding, or dominant, direction; and the flux is
formulated along the dominant direction. The dominant
direction is chosen, as described above, to be a physically
important direction, such as normal to a shock front and
not necessarily normal to a cell face. The computation of
this flux, therefore, includes two changes with respect to the
grid-aligned computation. The velocity components are
now in the new reference frame, and the left and right states
Rotated
FIG.
207
SOLVER
FIG. 8.
the MUSCL
Riemann
Template
scheme.
Solver
for cell update
MUSCL
Scheme
for the rotated
Riemann
solver
and
208
LEVY, POWELL,
are interpolated fro& the values in the nearest cell centers,
as described in the previous section.
Next, a flux normal to this direction, e.g., along a shock
front, must also be calculated:
F, = LU-J~M,, U,,h
UD,, U,wL, U,,)
=F,
COS(~--4)-
F,
states as in the pure rotation schemes. The minor flux is
calculated using certain cell average values, where the
specific cells are chosen based on the upwinding angle, It is
the flux functions which are interpolated rather than the
state quantities.
(23)
where the subscript M indicates that the flux is found in the
direction normal to the upwinding angle, or minor direction, and the flux is formulated along the minor direction. It
is not a requirement that the minor flux be calculated with
the same approximate Riemann solver as the dominant flux.
Even simple averaging could s&ice. In the final scheme, this
would lead to central differencing of the minor flux component. This approach is analogous to that used by Jameson
[ 31 for the potential equation, in which upwind differencing
is used for the streamwise derivatives and central differencing is used for the normal derivatives. It is possible to use
upwind differencing for both flux components. This would
lead to a better approximation of the effect of pressure
waves in the minor direction. This, of course, almost
doubles the computational effort of calculating the flux
through the cell face, relative to a rotated Riemann solver
which uses central differencing for the minor flux component. If the dominant direction is the streamline angle, then
only the two acoustic waves are required for the computation of the minor flux. If a different direction is used,
however, the full Riemann problem must be solved.’ In
the work of Goorjian [S, 61, which adopts the intrinsic
coordinate frame, only the acoustic waves are calculated.
The flux normal to the cell face is constructed by rotating
the above fluxes back to the coordinate frame normal to the
cell face:
WJ,,,
AND VAN LEER
sin(@--4).
(24)
Here, 6’ is the upwinding angle and 4 is the angle normal to
the cell face. The computation of the upwind biased fluxes in
a rotated frame, followed by rotating back to the computational frame, makes a difference only for the nonlinear
waves. For passive convection, such as entropy convection,
there is no change with respect to grid-aligned fluxes.
In the formulation of the characteristic upwinding scheme
for the scalar convection equation, the minor flux component is zero because the convection speed in that direction
is zero. It should also be noted that the minor flux component also exists in the conventional, grid-aligned formulation, but does not contribute to the flux through the cell
face.
In the earlier work of Davis [93, flux calculation and
interpolation are coupled together. The dominant flux is
calculated in the rotated frame using unbiased cell average
4. RESULTS
AND
DISCUSSION
In this section, the grid-aligned and rotated-Riemannsolver schemes are used to solve three Euler-flow test
geometries and conditions. These are:
1. Fifteen-degree wedge channel flow;
2. Shear wave oblique to the grid;
3. NACA 0012 airfoil.
In the first case, the effects of upwinding
minor flux formula are investigated.
direction
and
Fifteen Degree- Wedge Channel Flow. The geometry for
the first test case is a two-dimensional channel with a 15”
wedge on the lower wall. A 15” expansion corner is also
included to study expansion waves and wave interactions.
The inflow Mach number equals two. An oblique shock
wave is produced by the wedge and is somewhat weakened
by the expansion wave. The shock reflects from the upper
wall and passes through the rest of the expansion wave. The
expansion wave also reflects from the upper wall slightly
downstream of the shock-reflection point.
Conventional, analytical calculations predict a postshock Mach number of 1.454. For this Mach number, any
turning angle greater than 10.5” is too large to allow a
regular reflection off the upper wall. Thus, a “Mach reflection” results, with subsonic flow behind the Mach stem and
a slip surface trailing from the intersection of the incident,
normal, and reflected shock waves [25]. In this test case,
however, the incident shock wave interacts with the expansion wave. The shock incident on the upper wall may still be
too strong to allow a regular reflection; the numerical solutions seem to support this conclusion. The Mach number
along the lower wall, after the expansion corner, is predicted
to be 1.970.
The grid used is algebraic, with constant dx and
AY = (YInax- vmi,,)/N, where N is the number of cells in the
y direction, as shown in Fig. 9. The first-order grid-aligned
upwinding results are shown first in Fig. 10. The shock wave
is substantially diffused after reflection, as expected from a
first-order-scheme.
Figure 11 shows the result of using the local flow direction as the upwinding direction (streamwise upwinding).
The minor fluxes are computed with simple averaging,
which leads to central differencing for the minor flux component. Residual smoothing is required to stabilize the
scheme. The expansion wave and the reflected shock wave
ROTATED
RIEMANN
SOLVER
FIG. 9. Algebraic, 96 x 32 cell computational
are better resolved. This is especially true of the shock wave
reflecting from the lower wall near the exit of the channel.
The Mach contours of Fig. 11, and specifically the subsonic
Mach levels after the reflected shock (MMVIIN= 0.9573), hint
at the presence of the analytically predicted Mach reflection.
This feature is not present in the first-order grid-aligned
upwinding case.
When the flow angle is used for the dominant direction,
the only waves traveling in the minor direction are acoustic.
It is interesting that central-differencing
is stable at
all--even without added artificial dissipation.
One reason for the improved resolution and reduced dissipation is that the grid-independent scheme calculates a
part of the flux with central-differencing, which is a more
accurate approximation (am)
than upwind differencing
(I).
In fact, the flux through faces aligned with the local
flow direction is entirely composed of the central-differencing component. In the present scheme, the stabilizing effect
of using upwind differencing in the dominant direction
allows the use of central differencing in the cross direction
without artificial viscosity.
The other angle tested is the velocity magnitude, or q,
gradient angle. The q-gradient upwinding, as seen from
Fig. 12, shows substantial improvement in the resolution of
the shock waves, even over the streamwise upwinding. This
MIN = 1.2973
MAX
= 2.0000
209
grid for the 15” wedge.
should be expected, since now the upwinding direction is
normal to shock waves. However, the quality of the solution
has degraded in other respects. Overshoots of larger
magnitude have appeared in front of the shock, although
they are still small. There are noticeable “wiggles” in the
contour lines as well, especially in the contours of the
expansion fan ahead of the shock reflections.
To avoid noise in regions of uniform flow, a threshold
value for the q-gradient magnitude of 0.01 of the maximum
value in the domain is used. If the q-gradient magnitude
is greater than the threshold, then that angle is used.
Otherwise, the streamline angle is used. Figures 13 and 14
show the upwind directions for the streamwise and qgradient upwinding cases. For these figures, the data have
been thinned by a factor two, i.e., directions for every other
cell face are plotted. In the streamwise upwinding case, the
angles do not deviate much from the horizontal. The alignment normal to the incident and reflected shock waves is
quite evident in the q-gradient case.
Although
the rotated
Riemann
solvers perform
significantly better than the first-order grid-aligned method,
the higher-order, MUSCL, grid-aligned scheme still outperforms them all. This is shown in Fig. 1.5, for K = f with
Van Albada’s limiter. It has comparable shock resolution
without overshoots or “wiggles.”
lNc
= 0.0500
FIG. 10. Mach-number contours for the 15” wedge flow, first-order grid-aligned upwinding, 96 x 32 cell grid.
210
LEVY,
MIN
= 0.9573
POWELL,
MAX
AND
VA&LEER
mc
= 2.ooOl
= 0.0500
1.00
Y
0.00
-1.00
0.00
1.00
2.00
X
FIG.
11.
Mach-number contours for the 15” wedge flow, first-order streamwise upwinding, central differencing for minor fluxes, 96 x 32 cell grid.
MIN
= 0.9613
MAX
= 2.0279
INC = 0.0500
1.00
Y
0.00
FIG.
-1.00
1.00
2.00
X
12. Mach-number contours for the 15” wedge flow, first-order q-gradient upwinding, central differencing for minor fluxes, 96 x 32 cell grid.
0.00
FIG. 13. Upwinding angles for the 15” wedge flow, streamwise upwinding, 96 x 32 cell grid, thinning factor: 2.
FIG. 14. Upwinding angles for the 15” wedge flow, q-gradient upwinding, 96 x 32 cell grid, thinning factor: 2.
ROTATED
MIN
= 0.9258
MAX
RIEMANN
211
SOLVER
= 2.0000
INC
= 0.0500
1.00
Y
0.00
-1.00
0.00
1.00
2.00
X
FIG.
Mach-number
15.
contours
MIN
for the 15” wedge
= 1.0124
MAX
flow, x = f, MUSCL,
= 2.0243
INC
grid-aligned
upwinding,
96 x 32 cell grid
= 0.0500
1 .oo
Y
0.00
-1.00
0.00
1.00
2.00
X
FIG.
16.
Mach-number
contours
for the 15” wedge
MIN
flow,
first-order
MAX
= 1.0807
q-gradient
= 2.0026
upwinding,
INC
Riemann
solver
for minor
fluxes,
96 x 32 cell grid
= 0.0500
1 .oo
Y
0.00
-1.00
0.00
1 .oo
2.00
X
FIG.
17.
Mach-number
contours
for the 15” wedge
MIN
flow,
= 2.1798
limited
MAX
first-order
q-gradient
= 8.7ooO
INC
upwinding,
Riemann
solver
for minor
fluxes,
96 x 32 cell grid
= 0.5ooo
1.00
Y
0.00
0.00
-1.00
1.00
2
m
X
FIG.
18.
Mach-number
contours
for the Mach
8.7/2.9,
a = 45”,
shear-wave
problem,
K = f grid-aligned
upwinding,
96 x 32 cell grid.
212
LEVY, POWELL,
1.00
MIN = 2.3806
AND VAN LEER
MAX = 8.7412
INC
= 0.5ooo
X
FIG. 19. Mach-number contours for the Mach 8.7/2.9, c[ = 45”, shear-wave problem, first-order limited q-gradient upwinding, 96 x 32 cell grid.
Of the rotated-Riemann-solver
schemes, it appears that
the q-gradient direction is the best choice. It is chosen as the
upwinding direction for all the remaining cases.
It is possible that the use of central differencing for the
minor fluxes is causing some of the wiggles and destabilizing
effects of the present method. In the results shown in Fig. 16,
Roe’s approximate Riemann solver (the same as used for the
dominant fluxes) is used to compute the minor fluxes. While
there is a slight reduction in the sharpness of the shocks, the
overall quality of the solution has improved, as most of the
wiggles are eliminated. For all the remaining cases, the Roe
solver is used to compute the minor fluxes.
In Section 2 it was shown that the characteristic interpolation scheme would not be monotone and a method was
shown which, through limiting of the upwinding angle,
resulted in a monotone scheme. Results for the first-order
q-gradient scheme with angle limiting are shown in Fig. 17.
The results are now essentially non-oscillatory, and the
FIG. 20. O-type, 128 x 32 cell computational grid for airfoil flow cases.
solution is still substantially improved over the first-order
grid-aligned results.
Although the directional upwinding shows improved
performance in comparison to the grid-aligned method, it
comes at a price. The non-linearities introduced by the
method cause convergence difficulties. To help remedy this,
the upwind angle is recalculated only at specific times: at
startup and at intervals of 200 time steps thereafter. In total
it is calculated five times.
Shear Wave Oblique to Grid. The next example tests the
schemes’ ability to resolve a pure shear wave. In this case
there is no physical steepening mechanism, as for shock
waves, so it better indicates the diffusive properties of a
scheme. The flow is inclined at 45” with respect to the grid,
and the flow Mach numbers are 8.7 and 2.9 on the left and
right sides of the wave, respectively. Density is unity and
pressure is l/y on both sides of the wave. These conditions
are prescribed on the lower boundary, with the wave
originating at x = 0. The grid used is Cartesian.
Results for the K = + grid-aligned and the limited
q-gradient schemes are shown in Figs. 18 and 19. It is seen
that the q-gradient upwinding better represents the shear
wave than the K = f grid-aligned method in this case, Part of
the reason is the limitation of a cell-face reference frame. The
Riemann problems in a frame normal to the cell face will
indicate a shock or expansion wave, rather than the pure
shear wave that is actually present. The rotated Riemann
solver does not have this limitation, so its performance is
excellent on this problem.
NACA 0012 Airfoil Case. The previous channel flow
cases tested the various schemes on Cartesian or nearCartesian grids. For the airfoil test cases, a conventional
O-type grid is used, shown in Fig. 20. The freestream Mach
number is 1.20, and the angle of attack is zero. A “bow”
shock wave exists in front of the leading edge and “fishtail”
shocks emanate from the trailing edge. Results are shown
in Fig. 21. The bow and fishtail shock waves are better
resolved by the q-gradient schemes than by the first-order
ROTATED
a
MIN
= 0.0360
MAX
= 1.4841
INC
RIEMANN
= 0.0500
213
SOLVER
MIN
b
= 0.0120
MAX
= 1.5013
INC
= 0.0500
1.50
-1.50
-1.00
0.00
1.00
2.00
I.‘00
2
-
-700
2
-
MIN
C
= 0.0041
MAX
= 1.5012
INC = 0.0500
1.50
-1.50
= 0.0084
MAX
= 1.5215
INC
= 0.0500
1.50
-1.50
-1.00
0.00
1.00
2.00
FIG. 21. Mach-number contours for the NACA 0012 airfoil, M,
q-gradient; (c) first-order limited q-gradient; (d) K = f grid-aligned.
0.00
1 .oo
2.00
= 1.20, a = O.OO”,128 x 32 cell grid. (a) First-order grid-aligned; (b) first-order full
grid-aligned scheme. Here, the shocks are oblique to the
grid, so an improvement is expected. However, the K = f
grid-aligned scheme still gives the best results. The overshoot ahead of the bow shock in the full q-gradient scheme
is eliminated by the angle limiting.
AND
-1.00
2
5
5. CONCLUSIONS
MIN
d
RECOMMENDATIONS
In the present research, upwind schemes are formulated
for the two-dimensional Euler equations with the direction
of upwind differencing derived from strong flow features
instead of grid geometry. While multiple strong waves may
exist locally in the solution--each with its own optimum
upwind direction-a
single, dominant, upwinding angle is
used, and the resulting improvements are investigated. The
gradient of the velocity magnitude, q, is chosen as the best
direction because it is normal to both shock and shear
waves. To obtain input data for the numerical flux function,
an interpolation scheme is used to calculate states biased by
the chosen upwinding direction. The usual Riemann solver
is formulated in the upwinding direction, and an upwindbiased flux is computed in this direction. To complete the
flux through the cell face, a Riemann problem is also solved
in the direction normal to the q-gradient. The upwinding
angles are limited to avoid numerical oscillations; the
limiting
procedure
is based on an analysis of scalar convection schemes.
Several test problems are solved, including channel flows
and flows about airfoils. The limiting procedure derived for
the scalar convection problem is very effective when applied
to the rotated Riemann solver, almost completely eliminating overshoots. As with the scalar equation, the rotated
Riemann solver generally gives results better than a lirstorder grid-aligned method, but slightly inferior to a highorder, grid-aligned, MUSCL scheme. On the other hand, its
214
LEVY,
POWELL,
stencil is slightly more compact than that of the MUSCL
scheme. The solution to the pure shear-wave problem is the
one case where the rotated Riemann solver gives results
superior to even that of the best grid-aligned scheme. The
rotated Riemann solver finds the reference frame with zero
normal velocity, thereby reducing unnecessary dissipation
to the minimum possible. This feature would be important
in solutions to the Navier-Stokes equations.
Even in cases when it works well, the rotated-Riemannsolver approach still has a great weakness-its expense. The
rotated Riemann solver requires a significant amount of
additional memory per cell over grid-aligned methods, adds
computational effort per iteration and slows convergence.
The slower convergence is in part due to the extra
nonlinearity of the scheme, embodied in the local rotation
angle and limiter.
Despite its problems, the fundamental attraction of the
rotated Riemann solver still stands, namely, its respect for
the physics of the flow solution. The dramatic improvement
in accuracy-compare
Figs. 10 and 17-fully justifies the
search for grid-independent schemes. Future research may
best be concentrated on ways to use the rotated Riemann
solver in a selective manner, where it gives superior results,
and use high-order grid-aligned upwinding elsewhere. In
this manner some of the extra expense may be avoided.
Further reduction in expense may be realized if a more
efficient interpolation scheme can be developed.
Finally, it must be questioned whether the use of a single,
dominant upwind direction is sufficient. In the present
research, many inherently one-dimensional ideas are still
used even though the upwind direction is based on flow
solution features rather than the computational grid. For
further improvement to be obtained, a more “systemupwinded” approach may be required, where each state
variable may have its own optimum upwind direction,
subject to its own limiter.
ACKNOWLEDGMENTS
This work was funded
under a grant monitored
Science Foundation
under
Lea.
in part by the McDonnell
Aircraft
Company
by Dr. August
Verhoff,
and by the National
Grant EET-8857500,
monitored
by Dr. George
AND
VAN
LEER
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